ABSTRACT
This chapter presents an introduction to Hamiltonian T-spaces. It focuses on cobordisms and cohomological invariants; this material will appear in a joint publication with Victor Guillemin and Viktor Ginzburg. The chapter outlines joint work with Susan Tolman. Torus actions are restricted because they are easier and because many questions on actions of compact Lie groups reduce to questions on the actions of their maximal tori. The chapter provides a list of three excuses for studying Hamiltonian group actions: physics, representation theory and topology, geometry. The most useful source of examples of Hamiltonian torus actions is Delzant’s classification of the compact Hamiltonian T-spaces for which the dimension of the torus T is half the dimension of the manifold. Every Delzant space admits a complex structure which makes it into a Kahler toric variety. The image of the moment map, which is a convex polytope by the convexity theorem of Atiyah-Guillemin-Sternberg, determines the space up to isomorphism.
